Compositional (km,kn)-Shuffle Conjectures
CCOMPOSITIONAL ( km, kn ) –SHUFFLE CONJECTURES F. BERGERON, A. GARSIA, E. LEVEN, AND G. XIN
Abstract.
In 2008, Haglund, Morse and Zabrocki [16] formulated a Compositional formof the Shuffle Conjecture of Haglund et al. [15]. In very recent work, Gorsky and Negut bycombining their discoveries [14], [19] and [20], with the work of Schiffmann-Vasserot [22] and[23] on the symmetric function side and the work of Hikita [17] and Gorsky-Mazin [13] onthe combinatorial side, were led to formulate an infinite family of conjectures that extendthe original Shuffle Conjecture of [15]. In fact, they formulated one conjecture for each pair( m, n ) of coprime integers. This work of Gorsky-Negut leads naturally to the question asto where the Compositional Shuffle Conjecture of Haglund-Morse-Zabrocki fits into theserecent developments. Our discovery here is that there is a compositional extension of theGorsky-Negut Shuffle Conjecture for each pair ( km, kn ), with ( m, n ) co-prime and k > Contents
Introduction 11. The previous shuffle conjectures 22. The coprime case 53. Our Compositional ( km, kn )-Shuffle Conjectures 94. Symmetric function basics, and necessary operators 115. The SL [ Z ]-action and the Q operators indexed by pairs ( km, kn ) 156. The combinatorial side, further extensions and conjectures. 20References 32 Introduction
The subject of the present investigation has its origin, circa 1990, in a effort to obtaina representation theoretical setting for the Macdonald q, t -Kotska coefficients. This effortculminated in Haimain’s proof, circa 2000, of the n ! conjecture (see [7]) by means of theAlgebraic Geometry of the Hilbert Scheme. In the 1990’s, a concerted effort by many re-searchers led to a variety of conjectures tying the theory of Macdonald Polynomials to therepresentation theory of Diagonal Harmonics and the combinatorics of parking functions.More recently, this subject has been literally flooded with connections with other areas ofmathematics such as: the Elliptic Hall Algebra of Shiffmann-Vasserot, the Algebraic Ge-ometry of Springer Fibers of Hikita, the Double Affine Hecke Algebras of Cherednik, theHOMFLY polynomials, and the truly fascinating Shuffle Algebra of symmetric functions.This has brought to the fore a variety of symmetric function operators with close connec-tion to the extended notion of rational parking functions . The present work results froman ongoing effort to express and deal with these new developments in a language that is Date : July 6, 2014. This work was supported by NSERC, NSF grant DGE 1144086 and NSFC(11171231). a r X i v : . [ m a t h . C O ] J u l ore accessible to the algebraic combinatorial audience. This area of investigation involvesmany aspects of symmetric function theory, including a central role played by Macdonaldpolynomials, as well as some of their closely related symmetric function operators. One ofthe alluring characteristic features of these operators is that they appear to control in arather surprising manner combinatorial properties of rational parking functions. A close in-vestigation of these connections led us to a variety of new discoveries and conjectures in thisarea which in turn should open up a variety of open problems in Algebraic Combinatoricsas well as in the above mentioned areas.1. The previous shuffle conjectures
We begin by reviewing the statement of the Shuffle Conjecture of Haglund et al. (see[15]). In Figure 1 we have an example of two convenient ways to represent a parking function:a two-line array and a tableau. The tableau on the right is constructed by first choosing (cid:20) (cid:21) ⇐⇒ Figure 1.
Two representations of a parking functiona Dyck path. Recall that this is a path in the n × n lattice square that goes from (0 , n, n ) by north and east steps, always remaining weakly above the main diagonal (theshaded cells). The lattice cells adjacent and to the east of north steps are filled with cars , , . . . , n in a column-increasing manner. The numbers on the top of the two-line array arethe cars as we read them by rows, from bottom to top. The numbers on the bottom of thetwo line array are the area numbers, which are obtained by successively counting the numberof lattice cells between a north step and the main diagonal. All the necessary statistics of aparking function π can be immediately obtained from the corresponding two line array π := (cid:104) v v · · · v n u u · · · u n (cid:105) . To begin we letarea( π ) := n (cid:88) i =1 u i , anddinv( π ) := (cid:88) ≤ i
Our main actors on the symmetric function side are the operators D k and D ∗ k , introducedin [6], whose action on a symmetric function f [ x ] are defined by setting respectively D k f [ x ] := f [ x + M/z ] (cid:88) i ≥ ( − z ) i e i [ x ] (cid:12)(cid:12)(cid:12) z k , and (2.1) D ∗ k f [ x ] := f (cid:104) x − (cid:102) M /z (cid:105) (cid:88) i ≥ z i h i [ x ] (cid:12)(cid:12)(cid:12) z k . (2.2)with M := (1 − t )(1 − q ) and (cid:102) M := (1 − /t )(1 − /q ). The focus of the present work is thealgebra of symmetric function operators generated by the family { D k } k ≥ . Its connectionto the algebraic geometrical developments is that this algebra is a concrete realization of aportion of the Elliptic Hall Algebra studied Schiffmann and Vasserot in [21], [22], and [23].Our conjectures are expressed in terms of a family of operators Q a,b indexed by pairs ofpositive integers a, b . Here, and in the following, we use the notation Q km,kn , with ( m, n ) acoprime pair of non-negative integers and k an arbitrary positive integer. In other words, k is the greatest common divisor of a and b , and ( a, b ) = ( km, kn ). (3,5)(0,0) (2,3) Restricted to the coprime case, the definition of the operators Q m,n is first illustrated in a special case. For instance, to obtain Q , we startby drawing the 3 × , → (3 , a, b ) that is closest to and below the diagonal. In this case ( a, b ) = (2 , ,
5) = (2 ,
3) + (1 , Q , = 1 M (cid:2) Q , , Q , (cid:3) = 1 M (cid:0) Q , Q , − Q , Q , (cid:1) . (2.3) (2,3)(0,0) (1,1) We must next work precisely in the same way with the 2 × ,
3) =(1 ,
1) + (1 ,
2) and recursively set Q , = 1 M (cid:2) Q , , Q , (cid:3) = 1 M (cid:0) Q , Q , − Q , Q , (cid:1) . (2.4)Now, in this case, we are done, since it turns out that we can set Q ,k = D k . (2.5)In particular by combining (2.3), (2.4) and (2.5) we obtain Q , = 1 M ( D D D − D D D + D D D ) . (2.6) o give a precise general definition of the Q operators we use the following elementary numbertheoretical characterization of the closest lattice point ( a, b ) below the line (0 , → ( m, n ).We observe that by construction ( a, b ) is coprime. See [4] for a proof. Proposition 2.1.
For any pair of coprime integers m, n > there is a unique pair a, b satisfying the following three conditions (1) 1 ≤ a ≤ m − , (2) 1 ≤ b ≤ n − , (3) mb + 1 = na (2.7) In particular, setting ( c, d ) := ( m, n ) − ( a, b ) we will write, for m, n > , Split( m, n ) := ( a, b ) + ( c, d ) . (2.8) Otherwise, we set a ) Split(1 , n ) := (1 , n −
1) + (0 , , b ) Split( m,
1) := (1 ,
0) + ( m − , . (2.9)All pairs considered being coprime, we are now in a position to give the definition of theoperators Q m,n (restricted for the moment to the coprime case) that is most suitable in thepresent writing. Definition 2.1.
For any coprime pair ( m, n ) , we set Q m,n := (cid:40) M [ Q c,d , Q a,b ] if m > and Split( m, n ) = ( a, b ) + ( c, d ) , D n if m = 1 . (2.10)The combinatorial side of the upcoming conjecture is constructed in [17] by Hikita as theFrobenius characteristic of a bi-graded S n module whose precise definition is not needed inthis development. For our purposes it is sufficient to directly define the Hikita polynomial ,which we denote by H m,n [ x ; q, t ], using a process that closely follows our present renditionof the right hand side of (1.1). That is, we set H m,n [ x ; q, t ] := (cid:88) π ∈ Park m,n t area( π ) q dinv( π ) F ides( π ) [ x ] , (2.11)with suitable definitions for all the ingredients occurring in this formula. We will start withthe collection of ( m, n )- parking functions which we have denoted Park m,n . Again, a simpleexample will suffice.Figures 2 and 3 contain all the information needed to construct the polynomial H , [ x ; q, t ].The first object in Figure 2 is a 5 × , → (5 , lattice diagonal . Because of the coprimality of ( m, n ), the main diagonal, and any line par-allel to it, can touch at most a single lattice point inside the m × n lattice. Thus the maindiagonal (except for its end points) remains interior to the lattice cells that it touches. Sincethe path joining the centers of the touched cells has n − m − m + n − m − n − / m × n lattice that proceeds by north and east steps from (0 ,
0) to ( m, n ),always remaining weakly above the lattice diagonal, is said to be an ( m, n )-Dyck path. Forexample, the second object in Figure 2 is a (5 , a r m l e g Figure 2.
First combinatorial ingredients for the Hikita polynomial.path γ and the lattice diagonal is denoted area( γ ). In the third object of Figure 2, we havean (11 , leg and the arm of one of its cells (seeSection 4 for more details). Denoting by λ ( γ ) the Ferrers diagram above the path γ , wedefine dinv( γ ) := (cid:88) c ∈ λ ( γ ) χ (cid:18) arm( c )leg( c ) + 1 < mn < arm( c ) + 1leg( c ) (cid:19) . (2.12)As in the classical case an ( m, n )-parking function is the tableau obtained by labeling thecells east of and adjacent to the north steps of an ( m, n )-Dyck path with cars 1 , , . . . , n in a column-increasing manner. We denote by Park m,n the set of ( m, n )-parking functions.When ( m, n ) is a pair of coprime integers, it is easy to show that there are m n − such parkingfunctions. For more on the coprime case, see [1]. We will discuss further aspects of the moregeneral case in [4], a paper in preparation. Figure 3.
Last combinatorial ingredients for the Hikita polynomial.The first object in Figure 3 gives a (7 , × ranks . In the general case, this table is obtained by placing in the north-west corner of the m × n lattice a number of one’s choice. Here we have used 47 = ( m − n − − ut the choice is immaterial. We then fill the cells by subtracting n for each east step andadding m for each north step. Denoting by rank( i ) the rank of the cell that contains car i ,we define the temporary dinv of an ( m, n )-Parking function π to be the statistictdinv( π ) := (cid:88) ≤ i
Given any symmetric function f that is homogeneous of degree k , and anycoprime pair ( m, n ) , we proceed as follows Step 1: calculate the expansion f = (cid:88) λ (cid:96) k c λ ( q, t ) (cid:96) ( λ ) (cid:89) i =1 Q ,λ i , (3.1) Step 2: using the coefficients c λ ( q, t ) , set f km,kn := (cid:88) λ (cid:96) k c λ ( q, t ) (cid:96) ( λ ) (cid:89) i =1 Q mλ i ,nλ i . (3.2)Theoretical considerations reveal, and extensive computer experimentations confirm,that the operators that we should use to extend the rational parking function theory toall pairs ( km, kn ), are none other than the operators e km,kn obtained by taking f = e k in Al-gorithm 3.1. This led us to look for the construction of natural extensions of the definitionsof dinv( π ) and σ ( π ), that would ensure the validity of the following sequence of increasinglyrefined conjectures. The coarsest one of which is as follows. Conjecture 3.1.
For all coprime pair of positive integers ( m, n ) , and any k ∈ N , we have e km,kn · ( − ) k ( n +1) = (cid:88) π ∈ Park km,kn t area( π ) q dinv( π ) F ides( π ) , (3.3) o understand our first refinement, we focus on a special case. Inthe figure displayed on the right, we have depicted a 12 ×
20 lattice.The pair in this case has a gcd of 4. Thus here ( m, n ) = (3 ,
5) and k = 4. Note that in the general case, ( km, kn )-Dyck paths can hitthe diagonal in k − km × kn lattice square. Inthis case, in 3 places. We have depicted here a Dyck path which hitsthe diagonal in the first and third places.At this point the classical decomposition (discovered in [6]) e k = E ,k + E ,k + · · · + E k,k , (3.4)combined with extensive computer experimentations, suggested thatwe have the following refinement of Conjecture 3.1. Conjecture 3.2.
For all coprime pair of positive integers ( m, n ) , all k ∈ N , and if ≤ r ≤ k ,we have E rkm,kn · ( − ) k ( n +1) = (cid:88) π ∈ Park rkm,kn t area( π ) q dinv( π ) F ides( π ) , (3.5) where E rkm,kn is the operator obtained by setting f = E k,r in Algorithm 3.1. Here Park rkm,kn denotes the set of parking functions, in the km × kn lattice, whose Dyck path hit the diagonalin r places ( including (0 , . Clearly, (3.4) implies that Conjecture 3.1 follows from Conjecture 3.2. For example,the parking functions supported by the path in the above figure would be picked up by theoperator E × , × .Our ultimate refinement is suggested by the decomposition (proved in [16]) E k,r = (cid:88) α | = k C α C α · · · C α r · . (3.6)What emerges is the following most general conjecture that clearly subsumes our two previousconjectures, as well as Conjectures 1.1, 1.2, and 2.1. Conjecture 3.3 (Compositional ( km, kn )-Shuffle Conjecture) . For all compositions α =( a , a , . . . , a r ) | = k we have C ( α ) km,kn · ( − ) k ( n +1) = (cid:88) π ∈ Park ( α ) km,kn t area( π ) q dinv( π ) F ides( π ) , (3.7) where C ( α ) km,kn is the operator obtained by setting f = C α · in Algorithm 3.1 and Park ( α ) km,kn denotes the collection of parking functions in the km × kn lattice whose Dyck path hits thediagonal according to the composition α . For example, the parking functions supported by the path in the above figure would bepicked up by the operator C (1 , , × , × . We will later see that an analogous conjecture may bestated for the operator B ( α ) km,kn obtained by taking f = B α · in our general Algorithm 3.1,with α any composition of k . e will make extensive use in the sequel of a collection of results stated and perhapseven proved in the works of Schiffmann and Vasserot. Unfortunately most of this materialis written in a language that is nearly inaccessible to most practitioners of Algebraic Com-binatorics. We were fortunate that the two young researchers E. Gorsky and A. Negut, in aperiod of several months, made us aware of some of the contents of the latter publicationsas well as the results in their papers ([14], [19] and [20]) in a language we could understand.The present developments are based on these results. Nevertheless, for sake of completenesswe have put together in [4] a purely Algebraic Combinatorial treatment of all the backgroundneeded here with proofs that use only the Macdonald polynomial “tool kit” derived in the90’s in [2], [3], [8] and[9], with some additional identities discovered in [10].The remainder of this paper is divided into three further sections. In the next sectionwe review some notation and recall some identities from Symmetric Function Theory, andour Macdonald polynomial tool kit. This done, we state some basic identities that will beinstrumental in extending the definition of the Q operators to the non-coprime case. In thefollowing section we describe how the modular group SL ( Z ) acts on the operators Q m,n and use this action to justify our definition of the operators Q km,kn . Elementary proofs thatjustify the uses we make of this action are given in [4]. Here we also show how these operatorscan be efficiently programmed on the computer. This done, we give a precise constructionof the operators C ( α ) km,kn and B ( α ) km,kn , and workout some examples. We also give a compellingargument which shows the inevitability of Conjecture 3.3. In the last section we completeour definitions for all the combinatorial ingredients occurring in the right hand sides of (3.3),(3.5) and (3.7). Finally, we derive some consequences of our conjectures and discuss somepossible further extensions.4. Symmetric function basics, and necessary operators
In dealing with symmetric function identities, especially those arising in the theory ofMacdonald Polynomials, it is convenient and often indispensable to use plethystic notation.This device has a straightforward definition which can be implemented almost verbatim inany computer algebra software. We simply set for any expression E = E ( t , t , . . . ) and anysymmetric function f f [ E ] := Q f ( p , p , . . . ) (cid:12)(cid:12)(cid:12) p k → E ( t k ,t k ,... ) , (4.1)where ( − ) (cid:12)(cid:12) p k → E ( t k ,t k ,... ) means that we replace each p k by E ( t k , t k , . . . ), for k ≥
1. Here Q f stands for the polynomial yielding the expansion of f in terms of the power basis. We saythat we have a plethystic substitution of E in f .The above definition of plethystic substitutions implicitly requires that p k [ − E ] = − p k [ E ],and we say that this is the plethystic minus sign rule. This notwithstanding, we also needto carry out “ordinary” changes of signs. To distinguish the later from the plethystic minussign, we obtain the ordinary sign change by multiplying our expressions by a new variable“ (cid:15) ” which, outside of the plethystic bracket, is replaced by −
1. Thus we have p k [ (cid:15)E ] = (cid:15) k p k [ E ] = ( − k p k [ E ] . n particular we see that, with this notation, for any expression E and any symmetric function f we have ( ωf )[ E ] = f [ − (cid:15)E ] , (4.2)where, as customary, “ ω ” denotes the involution that interchanges the elementary and homo-geneous symmetric function bases. Many symmetric function identities can be considerablysimplified by means of the Ω-notation, allied with plethystic calculations. For any expression E = E ( t , t , · · · ) setΩ[ E ] := exp (cid:32)(cid:88) k ≥ p k [ E ] k (cid:33) = exp (cid:32)(cid:88) k ≥ E ( t k , t k , · · · ) k (cid:33) . In particular, for x = x + x + · · · , we see thatΩ[ z x ] = (cid:88) m ≥ z m h m [ x ] (4.3)and for M = (1 − t )(1 − q ) we haveΩ[ − uM ] = (1 − u )(1 − qtu )(1 − tu )(1 − qu ) . (4.4) armcoarmleg coleg Drawing the cells of the Ferrers diagram of a partition µ asin [18], For a cell c in µ , (in symbols c ∈ µ ), we have parametersleg( c ), and arm( c ), which respectively give the number of cellsof µ strictly south of c , and strictly east of c . Likewise we haveparameters coleg( c ), and coarm( c ), which respectively give thenumber of cells of µ strictly north of c , and strictly west of c .This is illustrated in the adjacent figure for the partition that sitsabove a path.Denoting by µ (cid:48) the conjugate of µ , the basic ingredients we need to keep in mind hereare n ( µ ) := (cid:96) ( µ ) (cid:88) k =1 ( k − µ k , T µ := t n ( µ ) q n ( µ (cid:48) ) , M := (1 − t )(1 − q ) ,B µ ( q, t ) := (cid:88) c ∈ µ t coleg( c ) q coarm( c ) , Π µ ( q, t ) := (cid:89) c ∈ µc (cid:54) =(0 , (1 − t coleg( c ) q coarm ) , and w µ ( q, t ) := (cid:89) c ∈ µ ( q arm( c ) − t leg( c )+1 )( t leg( c ) − q arm( c )+1 )Let us recall that the Hall scalar product is defined by setting (cid:104) p λ , p µ (cid:105) := z µ χ ( λ = µ ) , where z µ gives the order of the stabilizer of a permutation with cycle structure µ . TheMacdonald polynomials we work with here are the unique ([7]) symmetric function basis (cid:101) H µ [ x ; q, t ] } µ which is upper-triangularly related (in dominance order) to the modified Schurbasis { s λ [ x t − ] } λ and satisfies the orthogonality condition (cid:68) (cid:101) H λ , (cid:101) H µ (cid:69) ∗ = χ ( λ = µ ) w µ ( q, t ) , (4.5)where (cid:104)− , −(cid:105) ∗ denotes a deformation of the Hall scalar product defined by setting (cid:104) p λ , p µ (cid:105) ∗ := ( − | µ |− (cid:96) ( µ ) (cid:89) i (1 − t µ i )(1 − q µ i ) z µ χ ( λ = µ ) . (4.6)We will use here the operator ∇ , introduced in [2], obtained by setting ∇ (cid:101) H µ [ x ; q, t ] = T µ (cid:101) H µ [ x ; q, t ] . (4.7)We also set, for any symmetric function f [ x ],∆ f (cid:101) H µ [ x ; q, t ] = f [ B µ ] (cid:101) H µ [ x ; q, t ] . (4.8)These families of operators were intensively studied in the 90 (cid:48) s (see [3] and [9]) where theygave rise to a variety of conjectures, some of which are still open to this date. In particularit is shown in [9] that the operators D k , D ∗ k , ∇ and the modified Macdonald polynomials (cid:101) H µ [ x ; q, t ] are related by the following identities.(i) D (cid:101) H µ = − D µ ( q, t ) (cid:101) H µ , (i) ∗ D ∗ (cid:101) H µ = − D µ (1 /q, /t ) (cid:101) H µ , (ii) D k e − e D k = M D k +1 , (ii) ∗ D ∗ k e − e D ∗ k = − (cid:102) M D ∗ k +1 , (iii) ∇ e ∇ − = − D , (iii) ∗ ∇ D ∗ ∇ − = e , (iv) ∇ − e ⊥ ∇ = M D − , (iv) ∗ ∇ − D ∗− ∇ = − (cid:102) M e ⊥ , (4.9)with e ⊥ denoting the Hall scalar product adjoint of multiplication by e , and D µ ( q, t ) = M B µ ( q, t ) − . (4.10)We should mention that recursive applications of (4.9) (ii) and (ii) ∗ give D k = 1 M k k (cid:88) i =0 (cid:18) kr (cid:19) ( − r e r D e k − r , and (4.11) D ∗ k = 1 (cid:102) M k k (cid:88) i =0 (cid:18) kr (cid:19) ( − k − r e r D ∗ e k − r . (4.12)For future use, it is convenient to setΦ k := ∇ D k ∇ − and (4.13)Ψ k := − ( qt ) − k ∇ D ∗ k ∇ − . (4.14)The following identities are then immediate consequences of identities (4.9). See [4] fordetails. Proposition 4.1.
The operators Φ k and Ψ k are uniquely determined by the recursions a) Φ k +1 = 1 M [ D , Φ k ] and b) Ψ k +1 = 1 M [Ψ k , D ] (4.15) ith initial conditions a) Φ = 1 M [ D , D ] and b) Ψ = − e . (4.16)Next, we must include the following fundamental identity, proved in [4]. Proposition 4.2.
For a, b ∈ Z with n = a + b > and any symmetric function f [ x ] , wehave M ( D a D ∗ b − D ∗ b D a ) f [ x ] = ( qt ) b qt − h n (cid:20) − qtqt x (cid:21) f [ x ] . (4.17)As a corollary we obtain the following. Proposition 4.3.
The operators Φ k and Ψ k , defined in (4.13) and (4.14) , satisfy the follow-ing identity when a, b are any positive integers with sum equal to n . M [Ψ b , Φ a ] = qtqt − ∇ h n (cid:20) − qtqt x (cid:21) ∇ − . (4.18) Proof.
The identity in (4.17) essentially says that under the given hypotheses the operator M ( D ∗ b D a − D a D ∗ b ) acts as multiplication by the symmetric function ( qt ) b qt − h n [(1 − qt ) x / ( qt )].Thus, with our notational conventions, (4.17) may be rewritten as − ( qt ) − b M ( D ∗ b D a − D a D ∗ b ) = qtqt − h n (cid:20) − qtqt x (cid:21) . Conjugating both sides by ∇ , and using (4.13) and (4.14), gives (4.18). (cid:3) In the sequel, we will need to keep in mind the following identity which expresses theaction of a sequence of D k operators on a symmetric function f [ x ]. Proposition 4.4.
For all composition α = ( a , a , . . . , a m ) we have D a m · · · D a D a f [ x ] = f [ x + (cid:80) mi =1 M/z i ] Ω[ − z X ] z α (cid:89) ≤ i It suffices to see what happens when we use (2.1) twice. D a D a f [ x ] = D a f (cid:104) x + Mz (cid:105) Ω[ − z x ] (cid:12)(cid:12)(cid:12) z a = f (cid:104) x + Mz + Mz (cid:105) Ω[ − z ( x + Mz )]Ω[ − z x ] (cid:12)(cid:12)(cid:12) z a z a = f (cid:104) x + Mz + Mz (cid:105) Ω[ − z x ] Ω[ − z x ] Ω[ − M z /z ] (cid:12)(cid:12)(cid:12) z a z a = f (cid:104) x + Mz + Mz (cid:105) Ω[ − ( z + z ) x ] z a z a Ω[ − M z /z ] (cid:12)(cid:12)(cid:12) z z , and the pattern of the general result clearly emerges. (cid:3) . The SL [ Z ] -action and the Q operators indexed by pairs ( km, kn )To extend the definition of the Q operators to any non-coprime pairs of indices we needto make use of the action of SL [ Z ] on the operators Q m,n . In [4], SL [ Z ] is shown to act onthe algebra generated by the D k operators by setting, for its generators N := (cid:20) (cid:21) and S := (cid:20) (cid:21) , (5.1) N ( D k D k · · · D k r ) = ∇ ( D k D k · · · D k r ) ∇ − , (5.2)and S ( D k D k · · · D k r ) = D k +1 D k +1 · · · D k r +1 . (5.3)It is easily seen that (5.2) is a well-defined action since any polynomial in the D k that actsby zero on symmetric functions has an image under N which also acts by zero. In [4], thesame property is shown to hold true for the action of S as defined by (5.3).Since D k = Q ,k , and thus S Q ,k = Q ,k +1 , it recursively follows that S Q m,n = Q m,n + m . (5.4)On the other hand, it turns out that the property N Q m,n = Q m + n,n , (5.5)is a consequence of the following general result proved in [4] Proposition 5.1. For any coprime pair m, n we have Q m + n,n = ∇ Q m,n ∇ − . (5.6) It then follows from (5.4) and (5.5) that for any (cid:104) a cb d (cid:105) ∈ SL [ Z ] , we have (cid:20) a cb d (cid:21) Q m,n = Q am + cn,bn + dn . (5.7)The following identity has a variety of consequences in the present development. Proposition 5.2. For any k ≥ we have Q k +1 ,k = Φ k and Q k − ,k = Ψ k . In particular, forall pairs a, b , of positive integers with sum equal to n , it follows that M [ Q b +1 ,b , Q a − ,a ] = qtqt − ∇ h n (cid:20) − qtqt x (cid:21) ∇ − . (5.8) Proof. In view of (4.13) and the second case of (2.10), the first equality is a special instanceof (5.6). To prove the second equality, by Proposition 4.1), we only need to show that theoperators Q k − ,k satisfy the same recursions and base cases as the Ψ k operators. To begin,note that since Split( k, k + 1) = (1 , 1) + ( k − , k ) it follows that Q k,k +1 = 1 M (cid:2) Q k − ,k , Q , (cid:3) = 1 M (cid:2) Q k − ,k , D (cid:3) , (5.9)which is (4.15b) for Q k,k +1 . However the base case is trivial since by definition Q , = − e .The identity in (5.9) is another way of stating (4.18). (cid:3) igure 4. The four splits of (12,8).Our first application is best illustrated by an example. In Figure 4 we have depicted k versions of the km × kn rectangle for the case k = 4 and ( m, n ) = (3 , , → (4 × , × 2) by choosing a closest latticepoint below the diagonal. Namely,(12 , 8) = (2 , 1) + (10 , 7) = (5 , 3) + (7 , 5) = (8 , 5) + (4 , 3) = (11 , 7) + (1 , . Now, it turns out that the corresponding four bracketings[ Q , , Q , ] , [ Q , , Q , ] , [ Q , , Q , ] , and [ Q , , Q , ] , give the same symmetric function operator. This is one of the consequences of the identity in(5.8). In fact, the reader should not have any difficulty checking that these four bracketingsare the images of the bracketings in (5.8) for n = 4 by (cid:2) (cid:3) . Therefore they must also givethe same symmetric function operator since our action of SL [ Z ] preserves all the identitiessatisfied by the D k operators.In the general case if Split( m, n ) = ( a, b ) + ( c, d ), the k ways are given by(( u − m + a, ( u − n + b ) + (( k − u ) m + c, ( k − u ) n + d ) , with u going from 1 to k .The definition of the Q operators in the non-coprime case, as well as some of theirremarkable properties, appear in the following result proved in [4]. Theorem 5.1. If Split( m, n ) = ( a, b ) + ( c, d ) then we may set, for k > and any ≤ u ≤ k , Q km,kn = 1 M (cid:2) Q ( k − u ) m + c, ( k − u ) n + d , Q ( u − m + a, ( u − n + b (cid:3) . Moreover, letting Γ := (cid:104) a cb d (cid:105) we also have a) Q k,k = qtqt − ∇ h k (cid:20) − qtqt x (cid:21) ∇ − , and b) Q km,kn = Γ Q k,k . In particular, it follows that for any fixed ( m, n ) the operators (cid:8) Q km,kn (cid:9) k ≥ form a commutingfamily. Notice Γ ∈ SL [ Z ] since (3) of (4.15) gives ad − bc = 1. n immediate consequence of Theorem 5.1 is a very efficient recursive algorithm forcomputing the action of the operators Q km,kn on a symmetric function f . Let us recall thatthe Lie derivative of an operator X by an operator Y , which we will denote δ Y X , is simplydefined by setting δ Y X = [ X, Y ] := XY − Y X . It follows, for instance, that δ Y X = [[ X, Y ] , Y ] , δ Y X = [[[ X, Y ] , Y ] , Y ] , . . . Now, our definition gives Q , = 1 M [ Q , , Q , ] = 1 M [ D , D ] , and Q , = 1 M [ Q , , Q , ] = 1 M [[ D , D ] , D ] , and by induction we obtain Q k, = 1 M k − δ k − D D . Thus the action of the matrix S gives Q k,k +1 = M k − δ k − D D . In conclusion we may write Q k,k = 1 M [ Q k − ,k , Q , ] = 1 M [ Q k − ,k , D ] = 1 M k − (cid:2) δ k − D D , D (cid:3) . This leads to the following recursive general construction of the operator Q u,v . Algorithm 5.1. Given a pair ( u, v ) of positive integers: If u = 1 then Q u,v := D v else if u < v then Q u,v := S Q u,v − u else if u > v , then Q u,v := N Q u − v,v else Q u,v := M u − (cid:2) δ u − D D , D (cid:3) . This assumes that S acts on a polynomial in the D operators by the replacement D k (cid:55)→ D k +1 ,and N acts by the replacement D k (cid:55)→ ( − k M k δ k D D . We are now finally in a position to validate our construction (see Algorithm 3.1) of theoperators F km,kn . To this end, for any partition λ = ( λ , λ , . . . , λ (cid:96) ), of length (cid:96) ( λ ) = (cid:96) , it isconvenient to set h λ [ x ; q, t ] = (cid:18) qtqt − (cid:19) (cid:96) (cid:96) (cid:89) i =1 h λ i (cid:20) − qtqt x (cid:21) . (5.10)Notice that the collection { h λ [ x ; q, t ] } λ is a symmetric function basis. Thus we may carry outstep one of Algorithm 3.1. It may be good to illustrate this in a special case. For instance,when f = e we proceed as follows. Note first that for any two expressions A, B we have h [ AB ] = (cid:88) λ (cid:96) h λ [ A ] m λ [ B ] . Letting A = x (1 − qt ) / ( qt ) and B = qt/ ( qt − 1) gives( − e [ x ] = h [ − x ] = (cid:88) λ (cid:96) h λ [ x ; q, t ] m λ (cid:20) qtqt − (cid:21) (cid:16) qt − qt (cid:17) (cid:96) ( λ ) . hus e m, n = − (cid:88) λ (cid:96) m λ (cid:20) qtqt − (cid:21) (cid:18) qt − qt (cid:19) (cid:96) ( λ ) (cid:96) ( λ ) (cid:89) i =1 Q mλ i ,nλ i . (5.11)Carrying this out gives e m, n = − qt [2] qt Q m,n − qt (2 + qt )[3] qt [2] qt Q m,n Q m, n − ( qt ) [3] qt Q m, n , where for convenience we have set [ a ] qt = (1 − ( qt ) a ) / (1 − qt ).To illustrate, by direct computer assisted calculation, we get e , · ( − ) = s [ x ] + ( q + qt + t + q + t ) s [ x ]+( q + q t + qt + t + qt ) s [ x ]+( q + q t + qt + t + qt ) s [ x ]+( q + q t + q t + qt + t + q + 2 q t + 2 qt + t + q + qt + t ) s [ x ]+( q + q t + q t + q t + qt + t + q + 2 q t + 2 q t + 2 qt + t + q t + qt ) s [ x ]+( q + q t + q t + q t + q t + qt + t + q t + q t + q t + qt + q t ) s [ x ]Conjecturally, the symmetric polynomial e km,kn · ( − ) k ( n +1) should be the Frobenius charac-teristic of a bi-graded S kn module. In particular the two expressions (cid:10) e , · ( − ) , e (cid:11) and (cid:104) e , · ( − ) , s (cid:105) should respectively give the Hilbert series of the corresponding S module and the Hilbertseries of its alternants. Our conjecture states that we should have (as in the case of the Figure 5. Dyck paths in the 3 × odule of Diagonal Harmonics) (cid:10) e , · ( − ) , e (cid:11) = (cid:88) π ∈ Park , t area( π ) q dinv( π ) , and (cid:104) e , · ( − ) , s (cid:105) = (cid:88) γ ∈D , t area( γ ) q dinv( γ ) , where the first sum is over all parking functions and the second is over all Dyck paths in the3 × q + q t + q t + q t + q t + qt + t +5 q + 6 q t + 6 q t + 6 q t + 6 qt + 5 t +14 q + 19 q t + 20 q t + 19 qt + 14 t +24 q + 38 q t + 38 qt + 24 t +25 q + 40 qt + 25 t + 16 q + 16 t + 5 . Setting first q = 1, we get t + 6 t + 21 t + 50 t + 90 t + 120 t + 90 , which evaluates to 378 at t = 1; the second polynomial q + q t + q t + q t + q t + qt + t + q t + q t + q t + qt + q t , (5.12)evaluates to 12, at q = t = 1. All this is beautifully confirmed on the combinatorial side.Indeed, there are 12 Dyck paths in the 3 × { , , . . . , } ,is indeed 378. One may carefully check that this coincidences still holds true when one takesinto account the statistics area and dinv.In the next section we give the construction of the parking function statistics that mustbe used to obtain the polynomial e , · ( − ) by purely combinatorial methods. Figure 5shows the result of a procedure that places a square in each lattice cell, above the path,that contributes a unit to the dinv of that path. Taking into account that the area is thenumber of lattice squares between the path and the lattice diagonal, the reader should haveno difficulty seeing that the polynomial in (5.12) is indeed the q, t -enumerator of the aboveDyck paths by dinv and area. Remark 5.1. In retrospect, our construction of the operators C ( α ) km,kn has a certain degree ofinevitability. In fact, since the multiplication operator qtqt − h k (cid:20) − qtqt x (cid:21) corresponds to the operator Q ,k , and since we must have Q k,k = ∇ Q ,k ∇ − = qtqt − ∇ h k (cid:20) − qtqt x (cid:21) ∇ − by Proposition 5.1, then it becomes natural to set f k,k := ∇ f ∇ − . or any symmetric function f homogeneous of degree k . Therefore using the matrix Γ ofTheorem 5.1, we obtain f km,kn = Γ f k,k (5.13) In particular, it follows (choosing f = e k ) that e k,k · = ∇ e k ∇ − · = ∇ e k . The expansion e k = (cid:80) α | = k C α · yields the decomposition e k,k = (cid:80) α | = k C ( α ) k,k , and (5.13) thenyields e km,kn = (cid:88) α | = k C ( α ) km,kn . (cid:3) The combinatorial side, further extensions and conjectures. Our construction of the parking function statistics in the non-coprime case closely followswhat we did in section 1 with appropriate modifications necessary to resolve conflicts thatdid not arise in the coprime case. For clarity we will present our definitions as a collectionof algorithms which can be directly implemented on a computer.The symmetric polynomials arising from the right hand sides of our conjectures mayalso be viewed as Frobenius characteristics of certain bi-graded S n modules. Indeed, theyare shown to be by [17] in the coprime case. Later in this section we will present someconjectures to this effect.As for Diagonal Harmonics (see [15]), all these Frobenius Characteristics are sums ofLLT polynomials. More precisely, a ( km, kn )-Dyck path γ may be represented by a vector u = ( u , u , . . . , u kn ) with u = 0 , and u i − ≤ u i ≤ ( i − mn , (6.1)for all 2 ≤ i ≤ kn . This given, we setLLT( m, n, k ; u ) := (cid:88) π ∈ Park( u ) t area( u ) q dinv( u )+tdinv( π ) − maxtdinv( u ) s pides( π ) [ x ] , (6.2)where Park( u ) denotes the collection of parking functions supported by the path correspond-ing to u . We also use here the Egge-Loehr-Warrington (see [5]) result and substitute theGessel fundamental by the Schur function indexed by pides( π ) (the descent composition ofthe inverse of the permutation σ ( π )). Here σ ( π ) and the other statistics used in (6.2) areconstructed according to the following algorithm. Algorithm 6.1. (1) Construct the collection Park( u ) of vectors v = ( v , v , . . . , v kn ) which are the permutations of , , . . . , kn that satisfy the conditions u i − = u i = ⇒ v i − < v i . (2) Compute the area of the path, that is the number of full cells between the path andthe main diagonal of the km × kn rectangle, by the formula area( u ) = ( kmkn − km − kn + k ) / − kn (cid:88) i =1 u i . Denoting by λ ( u ) the partition above the path, set dinv( u ) = (cid:88) c ∈ λ ( u ) χ (cid:18) arm( c )leg( c ) + 1 ≤ mn < arm( c ) + 1leg( c ) (cid:19) . (4) Define the rank of the i th north step by km ( i − − knu i + u i / ( km +1) and accordinglyuse this number as the rank of car v i , which we will denote as rank( v i ) . This given,we set, for π = (cid:104) uv (cid:105) tdinv( π ) = (cid:88) ≤ r Proposition 6.1. For any β = ( β , β , . . . , β k ) , we have B β · = (cid:88) α (cid:22) β q c ( α,β ) C α · (6.6) where α (cid:22) β here means that α is a refinement of the reverse of β . That is α = α ( k ) · · · α (2) α (1) with α ( i ) | = β i , and in that case c ( α, β ) = k (cid:88) i =1 ( i − (cid:96) ( α ( i ) ) . Using (6.6) we can easily derive the following. Proposition 6.2. Assuming that Conjecture 3.3 holds, then for all compositions β = ( β , β , . . . , β (cid:96) ) | = k we have B ( β ) km,kn · ( − ) k ( n +1) = (cid:88) α (cid:22) β q c ( α,β ) (cid:88) π ∈ Park ( α ) km,kn t area( π ) q dinv( π ) F ides( π ) . Here B ( β ) km,kn is the operator obtained by setting f = B β · in Algorithm 3.1 and, as before, Park ( α ) km,kn denotes the collection of parking functions in the km × kn lattice whose Dyck pathhits the diagonal according to the composition α . A natural question that arises next is what parking function interpretation may be givento the polynomials Q km,kn · ( − ) k ( n +1) . Our attempts to answer this question lead to a varietyof interesting identities. We start with a known identity and two new ones. Theorem 6.1. For all n , we have Q n,n +1 · ( − ) n = ∇ e n , and (6.7) Q n,n · ( − ) n = ( − qt ) − n ∇ ∆ e h n = ∆ e n − e n . (6.8) roof. Since Q n +1 ,n = ∇ Q ,n ∇ − and Q ,n = D n , then Q n,n +1 · ( − ) n = ∇ D n ∇ − · ( − ) n = ∇ D n · ( − ) n . Recalling that ∇ − · ( − ) n = ( − n and D n · = ( − n e n , this gives (6.7). The proof of (6.8)is a bit more laborious. We will obtain it below, by combining a few auxiliary identities. (cid:3) Proposition 6.3. For any monomial m and λ (cid:96) ns λ [1 − m ] = (cid:40) ( − m ) k (1 − m ) if λ = ( n − k, k ) for some 0 ≤ k ≤ n − , Proposition 6.4. For all n , qtqt − h n (cid:20) − qtqt x (cid:21) = − ( qt ) − n n − (cid:88) k =0 ( − qt ) k s n − k, k [ x ] . (6.10) Proof. The Cauchy formula gives qtqt − h n (cid:20) − qtqt x (cid:21) = ( qt ) − n qt − (cid:88) λ (cid:96) n s λ [ x ] s λ [1 − qt ]and (6.9) with m = qt proves (6.10). (cid:3) Observe that when we set qt = 1, the right hand side of (6.10) specializes to − p n . Proposition 6.5. For all n , ∆ e h n [ x ] = n − (cid:88) k =0 ( − qt ) k s n − k, k [ x ] . (6.11) Proof. By (4.8) and (4.9) ( i ) we have ∆ e = ( I − D ) /M . Thus, by (2.1) M ∆ e h n [ x ] = h n [ x ] − h n [ x + M/z ] Ω[ − z x ] (cid:12)(cid:12)(cid:12) z = − n (cid:88) k =1 ( − k h n − k [ x ] e k [ x ] h k [ M ]= − n − (cid:88) k =1 ( − k s n − k, k [ x ] h k [ M ] + n − (cid:88) k =0 ( − k s n − k, k [ x ] h k +1 [ M ]= n − (cid:88) k =1 ( − k s n − k, k [ x ] ( h k +1 [ M ] − h k [ M ]) + s n [ x ] M. his proves (6.11) since the Cauchy formula and (6.9) give h n [ M ] = n − (cid:88) k =0 ( − t ) k (1 − t )( − q ) k (1 − q )= M n − (cid:88) k =0 ( qt ) k . (cid:3) Now, it is shown in [9] that e n [ x ] and h n [ x ] have the expansions e n [ x ] = (cid:88) µ (cid:96) n M B µ ( q, t )Π µ ( q, t ) w µ (cid:101) H µ [ x ; q, t ] , and (6.12) h n [ x ] = ( − qt ) n − (cid:88) µ (cid:96) n M B µ (1 /q, /t ) Π µ ( q, t ) w µ (cid:101) H µ [ x ; q, t ] . (6.13)We are then ready to proceed with the rest of our proof. Proof. [Proof of Theorem 6.1 continued] The combination of (6.10), (6.11) and (6.12) gives qtqt − h n (cid:20) − qtqt x (cid:21) = − ( qt ) − n ∆ e h n (6.14)= ( − n (cid:88) µ (cid:96) n M B µ ( q, t ) B µ (1 /q, /t ) Π µ ( q, t ) w µ (cid:101) H µ [ x ; q, t ] (6.15)Now from Theorem 5.1 we derive that Q n,n · ( − ) n = qtqt − ∇ h n (cid:20) − qtqt x (cid:21) ∇ − · ( − ) n = ( − n qtqt − ∇ h n (cid:20) − qtqt x (cid:21) . Thus (6.14) gives Q n,n · ( − ) n = ( − qt ) − n ∇ ∆ e h n . This proves the first equality in (6.8). The second equality in (6.16) gives Q n,n · ( − ) n = (cid:88) µ (cid:96) n T µ M B µ ( q, t ) B µ (1 /q, /t ) Π µ ( q, t ) w µ (cid:101) H µ [ x ; q, t ] . (6.16)But it is not difficult to see that we may write (for any µ (cid:96) n ) T µ B µ (1 /q, /t ) = e n − [ B µ ( q, t )]and (6.16) becomes (using (6.12)) Q n,n · ( − ) n = ∆ e n − (cid:88) µ (cid:96) n M B µ ( q, t ) Π µ ( q, t ) w µ (cid:101) H µ [ x ; q, t ]= ∆ e n − e n . (cid:3) o obtain a combinatorial version of (6.8) we need some auxiliary facts. Proposition 6.6. For all positive integers a and b , we have C a B b · = C a e b [ x ] = ( − /q ) a − s a, b [ x ] − ( − /q ) a s a, b − [ x ] , (6.17) and (cid:88) β | = n − a C a C β · = ( − /q ) a − s a, n − a [ x ] − ( − /q ) a s a, n − a − [ x ] . (6.18) Proof. The equality in (6.4) gives the first equality in (6.17) and the equivalence of (6.17)to (6.18), for b = n − a . Using (1.3) and (6.9) we derive that C a e b [ x ] = ( − /q ) a − b (cid:88) r =0 e b − r [ x ]( − r h r [(1 − /q ) /z ] Ω[ z x ] (cid:12)(cid:12)(cid:12) z a = ( − /q ) a − (cid:32) e b [ x ] h a [ x ] + (1 − /q ) b (cid:88) r =1 ( − r e b − r [ x ] h r + a [ x ] (cid:33) = ( − /q ) a − (cid:16) s a, b [ x ] + s a +1 , b − [ x ] ++(1 − /q ) (cid:16)(cid:80) br =1 ( − r s r + a, b − r [ x ] − (cid:80) br =2 ( − r s r + a, b − r [ x ] (cid:17) (cid:17) = ( − /q ) a − (cid:0) s a, b [ x ] + s a +1 , b − [ x ] − (1 − /q ) s a, b − [ x ] (cid:1) = ( − /q ) a − s a, b [ x ] − ( − /q ) a s a, b − [ x ] . (cid:3) The following conjectured identity is well known and is also stated in [15]. We will deriveit from Conjecture 1.2 for sake of completeness. Theorem 6.2. Upon the validity of the Compositional Shuffle conjecture we have ∇ ( − q ) − a s a, n − a = (cid:88) π ∈ Park n, ≥ a t area( π ) q dinv( π ) F ides( σ ( π )) , (6.19) where the symbol Park n, ≥ a signifies that the sum is to be extended over the parking functionsin the n × n lattice whose Dyck path’s first return to the main diagonal occurs in a row y ≥ a . Proof. An application of ∇ to both sides of (6.18) yields (cid:88) β | = n − a ∇ C a C β · = ( q ) a − ∇ ( − a − s a, n − a − ( q ) a ∇ ( − a s a, n − a − . (6.20)Furthermore, (6.19) is an immediate consequence of the fact that the Compositional ShuffleConjecture implies (cid:88) β | = n − a ; ∇ C a C β · = (cid:88) π ∈ Park n, = a t area( π )) q dinv( π ) F ides( σ ( π )) , (6.21)where the symbol Park n, = a signifies that the sum is to be extended over the parking functionswhose Dyck path’s first return to the diagonal occurs exactly at row a . (cid:3) ll these findings lead us to the following surprising identity. Theorem 6.3. Let ret( π ) denote the first row where the supporting Dyck path of π hits thediagonal. We have, for all n ≥ , that Q n,n · ( − ) n = (cid:88) π ∈ Park n [ret( π )] t t area( π )) − ret( π )+1 q dinv( π ) F ides( σ ( π )) . (6.22) Proof. Combining (6.7) with (6.11) gives Q n,n · ( − ) n = ( − qt ) − n n − (cid:88) k =0 ∇ ( − qt ) k s n − k, k . (6.23)Now this may be rewritten as Q n,n · ( − ) n = ( − qt ) − n n (cid:88) a =1 ∇ ( − qt ) n − a s a, n − a = n (cid:88) a =1 ∇ ( − qt ) − a s a, n − a , (6.24)and (6.19) gives Q n,n · ( − ) n = n (cid:88) a =1 t − a (cid:88) π ∈ Park n t area( π )) q dinv( π ) F ides( σ ( π )) χ (ret( π ) ≥ a ) (6.25)= (cid:88) π ∈ Park n t area( π )) q dinv( π ) F ides( σ ( π )) n (cid:88) a =1 t − a χ ( a ≤ ret( π )) . (6.26)This proves (6.22). (cid:3) Remark 6.1. Extensive computer experiments have revealed that the following difference isSchur positive e km +1 ,kn · ( − ) − t d ( km,kn ) e km,kn · , (6.27) where d ( km, kn ) is the number of integral points between the diagonals for ( km + 1 , kn ) and ( km, kn ) . Assuming that m ≤ n for simplicity sake, this implies that the following differenceis also Schur positive e kn,kn · − t a ( km,kn ) e km,kn · , (6.28) with a ( km, kn ) equal to the area between the diagonal ( km, kn ) and the diagonal ( kn, kn ) .This suggests that there is a nice interpretation of t a ( km,kn ) e km,kn · as a new sub-module ofthe space of diagonal harmonic polynomials. We believe that we have a good candidate forthis submodule, at least in the coprime case. We terminate with some surprising observations concerning the so-called Rational ( q, t ) -Catalan . In the present notation, this remarkable generalization of the q, t -Catalan polyno-mial (see [6]) may be defined by setting, for a coprime pair ( m, n ) C m,n ( q, t ) := (cid:10) Q m,n · ( − ) n , e n (cid:11) ∗ . (6.29)It is shown in [20], by methods which are still beyond our reach, that this polynomial mayalso be obtained by the following identity. heorem 6.4 (A. Negut) . C m,n ( q, t ) = n (cid:89) i =1 − z i ) z a i ( m,n ) i n − (cid:89) i =1 − qtz i /z i +1 ) (cid:89) ≤ i Let T n be the set of all standard tableaux with labels , , . . . , n . For agiven T ∈ T n , we set w T ( k ) = q j − t i − if the label k of T is in the i -th row j -th column. Wealso denote by S T the substitution set { z k = w − T ( k ) : 1 ≤ k ≤ n } . (6.32) We have C m,n ( q, t ) = (cid:88) T ∈ T n N m,n [ z ; q, t ] n (cid:89) i =1 (1 − z i w T ( i )) (cid:12)(cid:12)(cid:12) S T , (6.33) where the sum ranges over all standard tableaux of size n , and the S T substitution shouldbe carried out in the iterative manner. That is, we successively do the substitution for z followed by cancellation, and then do the substitution for z followed by cancellation, and soon. Proof. For convenience, let us write f ( u ) for Ω[ − M u ] and b i for a n +1 − i ( m, n ) this gives N m,n [ z ; q, t ] = n (cid:89) i =1 − z i ) z b i i n (cid:89) i =2 − qtz i /z i − ) (cid:89) ≤ i 1, the factors containing z d are 1(1 − z d ) z b d d − qt z d /z d − )(1 − qt z d +1 /z d ) χ ( d Inner and outer corners of a partition.This given, the rational function object of the constant term becomes the followingproper rational function in z d (provided b d ≥ − z b d d (cid:81) ( i,j ) ∈ IC [sh( T )] (1 − t i q j z d )(1 − z d qt w T ( d − (cid:81) ( i,j ) ∈ OC [sh( T )] (1 − t i − q j − z d ) (cid:89) d 1) inthe denominator cancels with a factor in the numerator.Therefore, the only factors, in the denominator that contribute to the constant term are those of the form (1 − q j − t i − z d ), for ( i, j ) an outer corner of T . For each such ( i, j ),construct T (cid:48) by adding d to T at the cell ( i, j ). By the partial fraction algorithm in [24]. hus, by the partial fraction algorithm, we obtain N m,n [ z ; q, t ] d − (cid:89) i =1 (1 − z i w T ( i )) (cid:12)(cid:12)(cid:12) S T (cid:12)(cid:12)(cid:12) z d = (cid:88) T (cid:48) N m,n [ z ; q, t ] d (cid:89) i =1 (1 − z i w T (cid:48) ( i )) (cid:12)(cid:12)(cid:12) S T (cid:12)(cid:12)(cid:12) z d = wT (cid:48) ( d ) = (cid:88) T (cid:48) N m,n [ z ; q, t ] d (cid:89) i =1 (1 − z i w T (cid:48) ( i )) (cid:12)(cid:12)(cid:12) S T (cid:48) , where the sum ranges over all T (cid:48) obtained from T by adding d at one of its outer corners.Applying the above formula to all T of size d − 1, and using the induction hypothesis,we obtain: N m,n [ z ; q, t ] (cid:12)(cid:12)(cid:12) z ··· z d = (cid:88) T N m,n [ z ; q, t ] d − (cid:89) i =1 (1 − z i w T ( i )) (cid:12)(cid:12)(cid:12) S T (cid:12)(cid:12)(cid:12) z d = (cid:88) T (cid:88) T (cid:48) N m,n [ z ; q, t ] d (cid:89) i =1 (1 − z i w T (cid:48) ( i )) (cid:12)(cid:12)(cid:12) S T (cid:48) = (cid:88) T (cid:48) N m,n [ z ; q, t ] d (cid:89) i =1 (1 − z i w T (cid:48) ( i )) (cid:12)(cid:12)(cid:12) S T (cid:48) , where the final sum ranges over all T (cid:48) of size d . (cid:3) Remark 6.2. We can see that the above argument only needs b ≥ , and b i ≥ − for i = 2 , , . . . , n . Thus the equality of the right hand sides of (6.30) and (6.33) holds true alsoif the sequence { a i ( m, n ) } ni =1 is replaced by any of these sequences. In fact computer datareveals that the constant term in (6.30) yields a polynomial with positive integral coefficientsfor a variety of choices of ( b , b , . . . , b n ) replacing the sequence { a i ( m, n ) } ni =1 . Trying toinvestigate the nature of these sequences and the possible combinatorial interpretations ofthe resulting polynomial led to the following construction. Given a path γ in the m × n lattice rectangle, we define the monomial of γ by setting z γ := n (cid:89) j =1 z e j j , (6.34)where e j = e j ( γ ) gives the number of east steps taken by γ at height j . Note that, bythe nature of (6.34), we are tacitly assuming that the path takes no east steps at height0. Note also that if γ remains above the diagonal (0 , → ( m, n ) then for each east step( i − , j ) → ( i, j ) we must have i/j ≤ m/n . In particular for the path γ that remains closestto the diagonal (0 , → ( m, n ), the last east step at height j must be given by i = (cid:98) jm/n (cid:99) .Thus z γ = z m,n = n (cid:89) j =1 z (cid:98) jm/n (cid:99)−(cid:98) ( j − m/n (cid:99) j . We can easily see that the series Ω[ z ] = n (cid:89) j =1 − z j ay be viewed as the generating function of all monomials of paths with north and east stepsthat end at height n and start with a north step. We will refer to the later as the NE-paths .Our aim is to obtain a formula for the q -enumeration of the NE-paths in the m × n latticerectangle that remain weakly above a given NE-path γ .Notice that if z γ = z r z r · · · z r m , and z δ = z s z s · · · z s m . (6.35)Then δ remains weakly above γ if and only if s i ≥ r i for 1 ≤ i ≤ m. When this happens let us write δ ≥ γ . This given, let us set C γ ( t ) := (cid:88) δ ≥ γ t area( δ/γ ) , where for γ and δ as in (6.35), we let area( δ/γ ) denote the number of lattice cells between δ and γ . In particular, for δ as in (6.35), we havearea( δ/γ ) = m (cid:88) i =1 ( s i − r i ) . Now we have the following fact Proposition 6.8. For all path γ , we have Ω[ z ] z γ n − (cid:89) i =1 − tz i /z i +1 (cid:12)(cid:12)(cid:12) z z ··· z n = (cid:88) δ ≥ γ t area( δ/γ ) . (6.36) Proof. Notice that each Laurent monomial produced by expansion of the product n − (cid:89) i =1 − tz i /z i +1 = n − (cid:89) i =1 (cid:32) t z i z i +1 + (cid:18) t z i z i +1 (cid:19) + · · · (cid:33) (6.37)may be written in the form n − (cid:89) i =1 (cid:18) t z i z i +1 (cid:19) c i = z c (cid:81) n − i =2 z ( c i − c i − ) + i z c n − n (cid:81) n − i =2 z ( c i − c i − ) − i n − (cid:89) i =1 t c i = z a z a · · · z a (cid:96) z b z b · · · z b (cid:96) t (cid:80) n − i =1 c i , (6.38)with (cid:96) = c + n − (cid:88) i =2 ( c i − c i − ) + = n − (cid:88) i =2 ( c i − c i − ) − + c n − ,a r = min (cid:40) j : c + j (cid:88) i =2 ( c i − c i − ) + ≥ r (cid:41) , and b r = min (cid:40) j : j (cid:88) i =2 ( c i − c i − ) − ≥ r (cid:41) . ince j = a r forces c j > 0, we see that the equality c + j (cid:88) i =2 ( c i − c i − ) + = c j + j (cid:88) i =2 ( c i − c i − ) − yields that in (6.38) we must have a r < b r , for 1 ≤ r ≤ (cid:96). (6.39)Now for the ratio in (6.38) to contribute to the constant term in (6.36), it is necessary andsufficient that the reciprocal of this ratio should come out of the expansionΩ[ z ] z γ = 1 z r z r · · · z r m (cid:88) d i ≥ z d z d · · · z d n n . (6.40)That is for some d , d , . . . , d n we must have z r z r · · · z r m z d z d · · · z d n n = z a z a · · · z a (cid:96) z b z b · · · z b (cid:96) . (6.41)Notice that z a z a · · · z a (cid:96) and z b z b · · · z b (cid:96) have no factor in common, since from the secondexpression in (6.38) we derive that each variable z i can appear only in one of these twomonomials. Thus z a z a · · · z a (cid:96) divides z r z r · · · z r m and z b z b · · · z b (cid:96) divides z d z d · · · z d n n and in particular (cid:96) ≤ m . But this, together with the inequalities in (6.39) shows that wemust have z d z d · · · z d n n = z s z s · · · z s m , with s i ≥ r i for 1 ≤ i ≤ m. In other words z d z d · · · z d n n must be the monomial of a NE-path δ ≥ γ . Moreover from theidentity in (6.41) we derive thatarea( δ/γ ) = ( b − a ) + ( b − a ) + · · · + ( b (cid:96) − a (cid:96) ) = − (cid:96) (cid:88) i =1 a i + (cid:96) (cid:88) i =1 b i . Thus from the middle expression in 5.35 it follows thatarea( δ/γ ) = − c − n − (cid:88) i =2 i ( c i − c i − ) + + n − (cid:88) i =2 i ( c i − c i − ) − + nc n − = − c − (cid:16) n − (cid:88) i =2 i ( c i − c i − ) (cid:17) + c n − = − c − n − (cid:88) i =2 ic i + n − (cid:88) i =1 ( i + 1) c i + nc n − = n − (cid:88) i =1 c i , which is precisely the power of t contributed by the Laurent monomial in (6.38).Finally, suppose that δ is a NE-path weakly above γ as in (6.35). This given, let usweight each lattice cell with southeast corner ( a, b ) with the Laurent monomial tz b /z b +1 . hen it is easily seen that for each fixed column 1 ≤ i ≤ m , the product of the weights ofthe lattice cells lying between δ and γ is precisely t s i − r i x r i /x s i . Thus m (cid:89) i =1 t s i − r i x r i x s i = t area( δ/γ ) z γ z δ . Since the left hand side of this identity is in the form given in (6.38), we clearly see thatevery summand of C γ ( t ) will come out of the constant term. (cid:3) Remark 6.3. It is easy to see that, for q = 1 , the constant term in (6.30) , reduces to theone in (6.36) with γ = γ (the closest path to the diagonal (0 , → ( m, n ) ). This is simplydue to the identity Ω[ − uM ] (cid:12)(cid:12)(cid:12) q =1 = (1 − u )(1 − qtu )(1 − tu )(1 − qu ) (cid:12)(cid:12)(cid:12) q =1 = 1 . 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E-mail address : [email protected] School of mathematical science, Capital Normal University, PR China E-mail address : [email protected]@gmail.com